Integral Representation Formulae

• Integral-representation formulae are explicit expressions that represent mathematical objects, such as functions or operators, as integrals over specialized kernels or domains.
• They employ methods like transformation techniques, contour integration, and projection methods to derive closed-form expressions and analytic continuations.
• Their applications span solving differential equations, numerical approximation, and analytic number theory, linking discrete sums with continuous analytic forms.

An integral-representation formula is an explicit expression of a mathematical object—typically a function, operator, or solution to an equation—as an integral, often over a nontrivial kernel or transformed domain. Such formulae play a fundamental role in analysis, special function theory, differential and integral equations, and applied mathematics, providing structural insight, closed-form computability, analytic continuation, and bridges between discrete and continuous realms.

1. Foundational Classes of Integral-Representation Formulae

Integral representations occur across a broad spectrum of contexts:

• Special Functions: Classical and generalized special functions admit integral forms interconnecting their analytic, algebraic, and asymptotic properties. Prototypical examples include Bessel, Legendre, Jacobi, and zeta functions. For example, the generalized k-Bessel function $W^k_{\nu,c}(x)$ possesses Sonine-type and duplication integral representations involving Beta kernels or generalized k-Bessel kernels (Mondal, 2016). Associated Legendre and Jacobi functions are expressed via both elementary and hypergeometric integral representations (Cohl et al., 11 Aug 2025, Cohl et al., 2013, Micheli, 2017).
• Operators and Solutions of Equations: Integral representations systematically encode solutions to ODE/PDE, integro-differential, delay, and difference equations. Notably, the Laplace and Fourier transform inversions—and their matrix and operator analogs—are realized as integral representations, with explicit operator-valued formulae mapping convolution, initial-value problems, and Wiener–Hopf systems to functional-integral transforms (Stenger, 2018, Peet, 2018, Baratchart et al., 2023).
• Analytic Number Theory and Summation Correction: Zeta, Dirichlet, and L-functions admit integral forms involving periodic corrections (Bernoulli kernels), yielding powerful analytic continuations and series for mathematical constants (Laville, 2013, Coffey, 2010).
• Information-Theoretic Quantities: Fractional moment and entropy calculations are reduced to low-dimensional integrals, often replacing high-dimensional convolution-type expectations with integral representations involving Laplace or Mellin-type kernels (Merhav et al., 2020).
• Integral Identities and Definite Integrals: Classical formulas—such as Cauchy-type, Euler-type, or generalized Mellin–Barnes representations—express combinations of algebraic or transcendental functions and finite series as explicit integrals, with generalizations encoding logarithmic, polynomial, or special-function corrections (Reynolds, 6 Aug 2024, Euler et al., 2011).

2. Key Methodologies in Derivation

Integral representations are constructed via several methodological paradigms:

• Transformation and Kernel Methods: Utilizing Mellin, Fourier, or Laplace transforms—often via inversion or Plancherel theory—enables recasting sums, operator equations, or spectral properties as integrals over kernel functions. For instance, the Fourier-transform approach is central to the representation of Bessel and Neumann series (Micheli, 2017).
• Contour Integration and Residue Calculus: For complex-variable integral representations, the evaluation of multivariate or contour integrals employs residue computation, analytic continuation, and deformation to compute generating kernels or to relate real-variable integrals to sums by Cauchy’s residue theorem (Constales et al., 2019, Euler et al., 2011, Reynolds, 6 Aug 2024).
• Change-of-Variable and Projection Techniques: These are used to connect analytic or combinatorial objects with their integral forms, such as the derivation of the determinant inverse via the integral over the product of linear forms on the complex torus, analytically projecting out invariant parts via group symmetry (Constales et al., 2019).
• Euler–Maclaurin and Periodic Bernoulli Kernels: Euler–Maclaurin formula, built using periodized Bernoulli polynomials, is foundational in representations correcting discrete sums to exact integrals by explicit periodic kernels, notably in zeta and polylogarithm function theory (Coffey, 2010).
• Contour and Mellin–Barnes Techniques in Definite Integrals: Expansion of classical definite integrals to include log-polynomial or parameter-dependent generalizations, as in the extended Cauchy integral, utilizes multi-contour techniques and Lerch transcendents or Hurwitz zeta function connections (Reynolds, 6 Aug 2024).
• Operator-Valued Functional Calculus: Indefinite integration, convolution, and transformation operators are given integral representations that, upon discretization or collocation (Legendre, Sinc methods), facilitate numerical computation and approximation theory (Stenger, 2018).

3. Specific Examples of Integral Representation in Research

Context / Object	Representative Integral Representation	Reference
Determinant Inverse in Small Neighborhood	$$\displaystyle\frac{1}{\det A}= \frac{1}{(2\pi i)^k} \int_{|z_r|=1} \prod_{r=1}^k \frac{w_r}{z_r}\,dz_r$$	(Constales et al., 2019)
Generalized k-Bessel Function	$$\displaystyle W^k_{\nu,c}(x) = \frac{1}{k\Gamma_k(\nu)} \int_0^1 t (1-t^2)^{\nu/k-1} W^k_{0,1,c}(xt)\,dt$$	(Mondal, 2016)
Dirac Delta Change-of-Variables Formula	$$\displaystyle\int_{\mathbb{R}^k} \delta(Ax)\,dx = \frac{1}{|\det A|}$$	(Constales et al., 2019)
Polylogarithm (with periodized Bernoulli kernel)	$$\displaystyle\operatorname{Li}_s(z) = \frac{z}{2} + \frac{z}{2^{s+1}} + \int_1^\infty (\ln z x^{s-1} - s x^{s-1}) z^x P_1(x)\,dx$$	(Coffey, 2010)
Dirichlet Series w/ Liouville Function Coefficient	$$\displaystyle\zeta_\lambda(s) = C(s) \int_0^\infty \mathcal{N}(x) x^{s-1/2}\,dx$$	(Laville, 2013)
PDEs (Boundary-Free PIE Representation)	$\mathcal{T}\dot{\hat{x}} = \mathcal{A}\hat{x}$, where $\mathcal{T},\mathcal{A}$ are 3-PI integral operators	(Peet, 2018)
Solutions to Non-Autonomous Delay Difference Eqns	$$\displaystyle y(t) = X(t,s)\varphi(0) + \int_{(s,\,t]} X(t,a)\,d\mu(a)$$	(Baratchart et al., 2023)
Associated Legendre/Ferrers Functions (multi-integral)	$\int_z^\infty\!\cdots\!\int_z^\infty \frac{\,_ν^{μ}(w)}{(w^2-1)^{\frac{μ}{2}}}\, (dw)^n = \frac{(-1)^n\,_ν^{\mu-n}(z)}{(-ν-μ)_n\,(z^2-1)^{(\mu-n)/2}}$	(Cohl et al., 2013)
Information-Theoretic Fractional Moment	$$\displaystyle\mathbb{E}[X^\rho] = 1 + \frac{\rho}{\Gamma(1-\rho)} \int_0^\infty \frac{e^{-u} - M_X(-u)}{u^{1+\rho}}\,du$$ (for $0<\rho<1$)	(Merhav et al., 2020)

4. Analytical Properties and Applications

Integral-representation formulae are characterized by several unifying analytical features:

• Convergence and Analytic Continuation: The structure of the kernel and the contour/domain of integration ensures absolute or conditional convergence in specific parameter regimes, and often analytically continues the function beyond its original domain.
• Symmetry and Invariance: Integral formulations are often constructed to respect group actions or invariance properties, for example, integrating over the complex torus or sphere projects out non-invariant or non-scalar parts (Constales et al., 2019, Cohl et al., 11 Aug 2025).
• Reduction to Series and Summation Identities: Expanding kernels or applying period-kernel corrections (as in the Euler–Maclaurin-based representations) allows conversion of integrals to rapidly convergent series, explicit in the theory of polylogarithms, zeta functions, and mathematical constants (Coffey, 2010).

Significant applications include:

• Spectral Analysis and PDE/ODE Solution Theory: Explicit integral representations underlie fundamental solution constructions, regularity and Liouville theorems (e.g., for $s$-harmonic functions in fractional Sobolev spaces) (Greco, 8 Jul 2024).
• Exact Evaluations, Asymptotics, and Numerical Methods: Integral representations are the starting point for asymptotic expansion via saddle point and stationary phase, for computation of special values, and for highly efficient numerical quadrature rules and approximation schemes (e.g., Sinc and Legendre collocation for operator inversion) (Stenger, 2018).
• Functional and Operator Calculus: Many integral representations extend directly (via the Dunford–Taylor or functional calculus methods) to define or invert functions of operators, as in the resolution of functional equations or inversion of transforms in both scalar and matrix/operator settings.

5. Integral Representations for Modern Mathematical Research

Integral-representation formulae have a central role in contemporary mathematical research, enabling:

• Scalable Algorithms and Symbolic Computation: As demonstrated in modern numerical analysis (e.g., PIETOOLS for PDE stability via integral operator representation (Peet, 2018)), and in algorithmic symbolic integration (systematic use of contiguous relations for indefinite integrals (Welz, 2012)), integral representations both underpin and improve the scalability and transparency of computational mathematics.
• Analytic Number Theory and Special Values: Representations with periodic correction kernels give analytic continuations and new perspectives on the depth of zeta, L-functions, and related conjectures; they also allow for fine-grained investigation of zeros, special values, and secondary terms in number-theoretic sums (Laville, 2013, Coffey, 2010).
• Unified Treatment of Discrete and Continuous Models: By deploying periodized kernels, Mellin transforms, and operator-valued calculus, integral representations systematically relate discrete sum evaluations to continuous analytic forms, blurring the traditional boundary between summation and integration as tools (Coffey, 2010, Reynolds, 6 Aug 2024).
• Mathematical Physics and Probability: In stochastic and quantum models, fractional moments, path integrals, and kernel representations facilitate both explicit calculation and probabilistic analysis via low-dimensional numerical integration (Merhav et al., 2020).

6. Historical Context and Generalizations

The concept predates the formalization of much analytic mathematics, with roots in the work of Euler, Cauchy, Dirichlet, and others. Cauchy’s original definite integrals and Euler’s reduction techniques were generalized and extended to incorporate log-polynomial factors and higher transcendental functions through Mellin–Barnes, Fourier, and other transform techniques (Reynolds, 6 Aug 2024, Euler et al., 2011).

Modern developments have universalized the framework—every family of orthogonal polynomials, special functions, and function solutions to linear equations has been equipped with explicit integral-representation formulae, and operator-theoretic and numerical methods have rendered these practical for analytic, algebraic, and computational purposes (Matos et al., 2017, Stenger, 2018).

Integral representations thus constitute a core infrastructure of advanced mathematical analysis, connecting classical and contemporary theory, computational science, and a multitude of application domains.